1. Field of the Invention
The present invention pertains to the art of seismic surveying for the exploration and production of petroleum reservoirs, and more specifically to the joint use of linear and rotational sensors on the free surface of the earth to enhance the spatial sampling of seismic wavefields.
2. Description of Relevant Art
There is a long term trend in seismic reflection surveying for oil and gas exploration and production to utilize sensing elements, commonly known as geophones, at decreasing spatial sample intervals. There is a continuing need for economical ability to measure seismic wavefields at finer spatial sampling.
It is well understood in many fields of physical science and engineering that a complete representation of mechanical motion requires the measurement of six degrees-of-freedom. Typically this is accomplished by measuring three orthogonal linear motions, and measuring rotations around three orthogonal axes.
There is a well established technology for measurement of the linear particle motion of seismic wavefields in the earth. Many commercial sensors exist to measure particle velocity or particle acceleration along one, or up to three, linear axes, utilizing various physical concepts to accomplish the measurements. It is most common to utilize measurements of the vertical particle motion.
There is an evolving commercial technology for measurement of the rotational particle motion of seismic wavefields in the earth. This includes sensors such as those commercially offered by, for example, MetTech (model Metr-3, http://www.mettechnology.com (2010)) in Jersey City, N.J. and Eentec (models R-1 and R-2, http://www.eentec.com/R-1_data_new.htm (2010)) in Kirkwood, Mo.
The utility of rotational seismic measurements is appreciated in earthquake and regional crustal seismology, as discussed, for example, in W. Lee, et. al. (eds.), “Rotational Seismology and Engineering Applications,” vol. 99 (no. 28 supp.) Bull. Seismological Society of America (May 2009). Seismic rotational motion is commonly understood to be the vector curl of the infinitesimal displacement field. The existing rotational sensors are understood to measure the components of this vector curl.
The significant effect of the free surface of the earth on stress fields, strain fields, and seismic wave fields is widely understood. This is described, for example, in K. Aki & P. Richards, Quantitive Seismology, University Science Books, pp 128ff, 184-185 (2002). The stress components, commonly referred to as σxz and σyz, involving the nominal vertical direction, normal to the free surface, have zero value at the free surface.
In the field of sampled data analysis, there is a well established technology for enhanced sampling rate by utilizing the sampling of the ordinate values and the slope of the function being sampled. This technology is commonly understood for time series data, and is also directly applicable to spatial sampling. This technology, often referred to as Ordinate and Slope sampling, is described, for example, in R. Bracewell, The Fourier Transform and its Applications, McGraw-Hill, pp. 230-232 (2000).